Can Blender manipulate remote assets, without me having to download them to the client machine manually?
Not directly possible, Blender needs its data to be local (it does not understand uri, FTP, etc.).

You could set up a 'fake' local file system on your students' machine, though, which would actually be a network one (like a samba/windows sharing, or other similar solution). It will work as long as Blender can see it as a local directory.